We study the time-dependent heat equation on its space-time domain that is discretised by a k-spacetree.k-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale r...We study the time-dependent heat equation on its space-time domain that is discretised by a k-spacetree.k-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint.The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation’s elliptic operator with a multiscale solution propagation in time.While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded,the holistic approach promises to exhibit a better parallel scalability than classical time stepping,adaptive dynamic refinement in space and time fall naturally into place,as well as the treatment of periodic boundary conditions of steady cycle systems,on-time computational steering is eased as the algorithm delivers guesses for the solution’s long-term behaviour immediately,and,finally,backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.展开更多
We couple different flow models,i.e.a finite element solver for the Navier-Stokes equations and a Lattice Boltzmann automaton,using the framework Peano as a common base.The new coupling strategy between the meso-and m...We couple different flow models,i.e.a finite element solver for the Navier-Stokes equations and a Lattice Boltzmann automaton,using the framework Peano as a common base.The new coupling strategy between the meso-and macroscopic solver is presented and validated in a 2D channel flow scenario.The results are in good agreement with theory and results obtained in similar works by Latt et al.In addition,the test scenarios show an improved stability of the coupledmethod compared to pure Lattice Boltzmann simulations.展开更多
基金supported by Award No.UKc0020,made by the King Abdullah University of Science and Technology(KAUST).
文摘We study the time-dependent heat equation on its space-time domain that is discretised by a k-spacetree.k-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint.The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation’s elliptic operator with a multiscale solution propagation in time.While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded,the holistic approach promises to exhibit a better parallel scalability than classical time stepping,adaptive dynamic refinement in space and time fall naturally into place,as well as the treatment of periodic boundary conditions of steady cycle systems,on-time computational steering is eased as the algorithm delivers guesses for the solution’s long-term behaviour immediately,and,finally,backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.
文摘We couple different flow models,i.e.a finite element solver for the Navier-Stokes equations and a Lattice Boltzmann automaton,using the framework Peano as a common base.The new coupling strategy between the meso-and macroscopic solver is presented and validated in a 2D channel flow scenario.The results are in good agreement with theory and results obtained in similar works by Latt et al.In addition,the test scenarios show an improved stability of the coupledmethod compared to pure Lattice Boltzmann simulations.
